## Perfect “Square” Circle

Numbers 1 to 32 are placed along the circumference of a circle without repeating any number and still the sum of any two adjacent numbers in this circle is a perfect square! ## Sum of Consecutive Cubes (Visual Proof)

The sum of the first n cubes is the square of the nth triangular number:
13 + 23 + 33 + 43 + 53 + . . . + n3 = (1 + 2 + 3 + 4 + 5 + . . . + n)2. ## Inverse Powers of Phi

Summation of Alternating Inverse Powers of Phi… ## Visual Proof (sum of cubes)

The sum of the sequence of the first n cubes equals [n(n+1)/2]² as shown below:
1³+2³+3³+…+n³ = (1+2+3+…+n)² = [n(n+1)/2]² ## Infinite Pythagorean Triplets

Consider the following simple progression of whole and fractional numbers (with odd denominators):
1 1/3, 2 2/5, 3 3/7, 4 4/9, 5 5/11, 6 6/13, 7 7/15, 8 8/17, 9 9/19, …
Any term of this progression can produce a Pythagorean triplet, for instance:
4 4/9 = 40/9; the numbers 40 and 9 are the sides of a right triangle, and the hypotenuse is one greater than the largest side (40 + 1 = 41). ## Cauchy Product

A clever visualization of a Cauchy Product.
Each black square has value 1, each red square has value -1. Read more

## Fibonacci Spiral Jigsaw Puzzle

Each piece of this puzzle is similar (the same shape at a different size). The placement of the pieces is based on the golden angle (≈137.5º), and results in a pattern frequently found in nature (phyllotaxis), for instance on sunflowers. The puzzle features 8 spirals in one direction, and 13 in the other. You can build your own Fibonacci spiral puzzle by following John Edmark’s tutorial. ## Circles and Golden Ratio

The last digit of the numbers in the Fibonacci Sequence are cyclic, they form a pattern that repeats after every 60th number: 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1.

## Surprising Limit

Amazingly, this sequence of fractions converges to 0.70710678118…, or to be precise, to √2/2. The sequence is related to the Prouhet-Thue-Morse sequence. 